Hello,
The authors are Verdier and Grothendieck, I doubt they made mistakes, and specially in the very basic definitions of the whole theory.
This made me smile, but I agree!
There is something odd here but I am not inclined to take time to clarify it, I will sit and wait to see if some in the list come out with an explanation.
I think the explanation is rather simple. The first point is that the foundations here are Bourbaki set theory: everything is a set, including a category. When they say that C belongs to U, they literally mean that C, when rigorously formalised as a set, is an element of U. The same goes for a functor. The second point is that U-smallness of a set X is defined to be a set which is isomorphic to an element of U, not necessarily an actual element of U. The word 'isomorphic' is underlined. Let us consider (C2) first. The axioms of a universe only allow one to construct sets in the universe from other sets in the universe. Since U is not in U, there is no way that any definition/construction involving it can produce a set in U. Certainly one needs to involve U to be able to define the Hom sets of Func(C, U-Ens). But the Hom sets of Func(C, U-Ens) are isomorphic to elements of U, i.e. are U-small. This is because U-Ens is a U-category, i.e. the Hom sets are U-small. One can make the same point about the set of objects of Func(C, U-Ens). One needs U to define it, so it is certainly not an element of U. It is not even isomorphic to an element of U, because this would imply that the cardinality of U is strictly less than U. In (C1), it is not exactly this that is asked, but rather that the set of objects of Func(C, U-Ens) is a subset of U. This is impossible for the same kind of reasons: one will need to consider for instance a subset of the set C x U to define it, and there is no way that show that this is a subset of U (it is perfectly possible if instead of U we have some element u of U, because then the product of C and u belongs to U). Hence Func(C, U-Ens) is, as SGA claims, the prototypical example to illustrate why the definition of a U-category is exactly the way it is. Best wishes, Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]